University Business Mathematics — Limits, Derivatives and Marginal Analysis Worksheet

11 of the 12 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one. 11 of the 12 questions are printed below. The other 1 is built on a diagram or a table of values that does not translate to the page, so it is in the free PDF — marked below where it would have come.

1. Q1Limits from Graphs and by Algebra

   This question is built around a diagram or a table of values. Open it in the PDF.

2. Q2Limits from Graphs and by Algebra

   Evaluate each limit exactly. Try direct substitution first, and show the algebra that removes a $\frac{0}{0}$ form wherever one appears.

   1. ${\displaystyle {lim}_{x\to 4}\frac{x^2-2x-8}{x^2-16}}$
   2. ${\displaystyle {lim}_{x\to 5}\frac{\sqrt{x+4}-3}{x-5}}$
   3. ${\displaystyle {lim}_{x\to 2}\frac{{\displaystyle \frac{1}{x}}-{\displaystyle \frac{1}{2}}}{x-2}}$
   4. ${\displaystyle {lim}_{x\to 3}\frac{x^2+1}{x+2}}$
3. Q3Infinite Limits, Limits at Infinity and Asymptotes

   A board-game publisher has fixed costs of $18 000 for a print run, plus $9 for each copy printed. Let $x>0$ be the number of copies.

   1. Write the average cost per copy, $\overline{C}(x)={\displaystyle \frac{C(x)}{x}}$, and find $\overline{C}(1000)$ and $\overline{C}(6000)$.
   2. Find ${\displaystyle {lim}_{x\to \infty }\overline{C}(x)}$ and ${\displaystyle {lim}_{x\to 0^{+}}\overline{C}(x)}$, and give the equations of the horizontal and vertical asymptotes of the graph of $\overline{C}$.
   3. Explain in one sentence why no print run, however large, brings the average cost down to $9.00 a copy, and find the run that brings it to $10.00.
4. Q4Infinite Limits, Limits at Infinity and Asymptotes

   Let $$g(x)=\frac{2x^2-18}{x^2+2x-15}.$$

   1. The denominator is zero at two values of $x$. For each, decide whether the graph of $g$ has a vertical asymptote or a hole there, and justify with a limit.
   2. At the vertical asymptote, find both one-sided limits.
   3. Find ${\displaystyle {lim}_{x\to \infty }g(x)}$ and ${\displaystyle {lim}_{x\to -\infty }g(x)}$, and state the horizontal asymptote.
   4. For $h(t)={\displaystyle \frac{900}{3+2e^{-0.5t}}}$, find ${\displaystyle {lim}_{t\to \infty }h(t)}$ and ${\displaystyle {lim}_{t\to -\infty }h(t)}$, and state both horizontal asymptotes of its graph.
5. Q5Continuity and Sign Charts

   Let $$f(x)=\frac{x^2-2x-8}{x-1}.$$

   1. On which intervals is $f$ continuous?
   2. Build a sign chart for $f$, and use it to solve $f(x)\le 0$. Give the answer in interval notation.
6. Q6Rates of Change and the Definition of the Derivative

   A coffee roaster's weekly cost of roasting $x$ kilograms of beans is $$C(x)=200+6x+0.01x^2\text{dollars}.$$

   1. Find the average rate of change of cost as production rises from $100$ kg to $150$ kg, with units.
   2. Simplify ${\displaystyle \frac{C(100+h)-C(100)}{h}}$ for $h\ne 0$.
   3. Let $h\to 0$ to find the instantaneous rate of change of cost at $100$ kg, and say in one sentence what it means.
7. Q7Rates of Change and the Definition of the Derivative

   A preserves maker finds that it can sell $x$ hundred jars of jam a month at a price of $$p=D(x)=\frac{180}{x+3}\text{dollars per jar},x\ge 0.$$

   1. Use the definition $D^{\prime }(x)={\displaystyle {lim}_{h\to 0}\frac{D(x+h)-D(x)}{h}}$ to find $D^{\prime }(x)$.
   2. Find $D(3)$ and $D^{\prime }(3)$, and interpret $D^{\prime }(3)$ with units.
8. Q8Power, Sum and Constant Multiple Rules

   Differentiate each function. Rewrite it first as a sum of constant multiples of powers of $x$ where that is needed, and give the answer without negative or fractional exponents in (b).

   1. $f(x)=4x^5-3x^2+7x-9$
   2. $g(x)=6\sqrt{x}-{\displaystyle \frac{8}{x^2}}+\sqrt[3]{x^2}$, for $x>0$
   3. $h(x)=(x^2+3{)}^2$
   4. $k(x)={\displaystyle \frac{3x^3-2x+5}{x}}$, for $x\ne 0$
9. Q9Power, Sum and Constant Multiple Rules

   Let $f(x)=2x^3-3x^2-36x+5$.

   1. Find every point on the graph of $f$ at which the tangent line is horizontal.
   2. Find the equation of the tangent line at $x=1$.
10. Q10Marginal Cost, Revenue and Profit

    A candle maker's weekly cost of making $x$ candles is $$C(x)=800+6x-0.004x^2\text{dollars},0\le x\le 600.$$

    1. Find the marginal cost function $C^{\prime }(x)$.
    2. Find $C^{\prime }(250)$ and interpret it.
    3. Find the exact cost of making the $251$st candle, and compare it with $C^{\prime }(250)$.
11. Q11Marginal Cost, Revenue and Profit

    A small workshop sells yoga mats. At a price of $p$ dollars it sells $x$ mats a week, where $p=120-0.04x$ for $0\le x\le 3000$. Its weekly cost is $C(x)=3000+40x$ dollars.

    1. Write the revenue $R(x)$ and the profit $P(x)$, and find the marginal revenue $R^{\prime }(x)$ and the marginal profit $P^{\prime }(x)$.
    2. Find $R^{\prime }(500)$, $P^{\prime }(800)$ and $P^{\prime }(1200)$, and interpret each.
12. Q12Synthesis — drawing on several topics in this unit

    A ski-rental shop can rent $x$ ski packages a week at a price of $p=80-0.1x$ dollars each, for $0\le x\le 800$. Its weekly cost is $C(x)=5000+20x$ dollars.

    1. Write the weekly profit $P(x)$.
    2. Use a sign chart to find the numbers of weekly rentals at which the shop makes a profit, that is, at which $P(x)>0$.
    3. Find the marginal profit $P^{\prime }(x)$, and use its sign to say for which numbers of rentals one more rental raises profit and for which it lowers it.

The 8 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Reading a limit off a graph with a jump

Q1 models a stock of cartons that runs down steadily and then jumps when a delivery lands. The graph has an open circle and a solid dot above the same day, and each mark answers a different question. The open circle is where one branch is heading; the solid dot is the value the function actually takes on that day.

Part (c) is a test of that rule and nothing else: compare the two numbers from part (b) and say which condition they meet or fail. Part (d) turns a left-hand limit into plain language for a manager — say what quantity it describes and at what moment, in the units of the problem.

0/0 is a signal, not an answer

Q2 asks for four limits and tells you the order of work: substitute first. If substitution gives a number, that number is the limit and you are done. If it gives 0/0, the expression is hiding a common factor, and the job is to expose and cancel it before substituting again.

Average cost and limits at infinity

Q3 builds a cost function from a fixed cost and a cost per copy, and asks for the average cost per copy, cost divided by quantity. Write C(x) first, then divide it by x and split the result into separate terms: one term that still contains x and one that does not.

For part (b), ask what each term does as x grows without bound, and again as x shrinks toward zero from the right. A constant divided by an ever-larger number shrinks toward zero; a positive constant divided by an ever-smaller positive number grows without bound. A limit at infinity that is a number gives a horizontal asymptote; an infinite limit at a finite x gives a vertical one. Part (c) wants a reason in one sentence that follows from part (b), then an equation: set the average cost equal to the target and solve for x.

Hole or vertical asymptote

Q4(a) gives a rational function whose denominator vanishes at two values of x and asks you to classify each. The decision comes from factoring numerator and denominator completely.

Justify each classification with a limit, as the question asks: at a hole the limit is a finite number after cancelling; at a vertical asymptote it grows without bound. For part (b), test the sign of the simplified expression just to the left and just to the right of the asymptote: the sign of each factor decides whether each side heads up or down.

For part (c), divide every term of the numerator and the denominator by the highest power of x in the denominator and let x → ∞, then x → −∞; every term with x left in a denominator fades to zero. Part (d) is a different shape: an exponential in the denominator. There, ask what e^(−0.5t) does as t → ∞ and what it does as t → −∞. The two behaviours are opposite, which is why the question asks for two horizontal asymptotes.

Continuity and sign charts

Q5(a) uses the fact that a rational function is continuous everywhere its denominator is not zero, so its intervals of continuity are read straight off the denominator. Q5(b) turns that fact into a method for inequalities.

Continuity is what makes the method honest: between two neighbouring cut points the function is continuous and never zero, so it cannot change sign there, and one test value speaks for the whole interval.

Average rate, then instantaneous rate

Q6 walks the definition of the derivative in three steps on a roaster's cost. Part (a) is the slope between two production levels, in dollars per kilogram. Part (b) is the same slope with the second point left general: expand C(100 + h), subtract C(100), and notice that every surviving term has a factor h, which is what lets you divide by h. Part (c) lets h → 0 in what remains.

The interpretation sentence carries the marks as much as the number: name the quantity, the production level, and the units — dollars per kilogram — and say that it describes how fast cost is rising at that level, not the cost itself.

Q7 applies the same definition to a demand function with x in the denominator, and asks for D′(x) as a function rather than at one point. Write D(x + h) − D(x) over a single common denominator, simplify the numerator until a factor h appears, cancel it, and only then let h → 0. Part (b) evaluates both D and D′ at one quantity; the interpretation must keep the units straight, since x counts hundreds of jars and p is dollars per jar.

The power, sum and constant multiple rules

Once the definition has been used, the rules take over. The power rule works for any real exponent, so the skill Q8 drills is rewriting before differentiating.

Q9 connects the derivative to the tangent line. A horizontal tangent is a point where f′(x) = 0: differentiate, set the derivative to zero, factor, and then substitute each x back into f, not into f′, to get the points. For part (b), the slope is f′(1), the point is (1, f(1)), and the line is y − f(1) = f′(1)(x − 1).

Marginal cost, revenue and profit

In business language, the derivative of cost is marginal cost, and it approximates the cost of producing one more unit. Q10 asks for the marginal cost function, its value at one production level, and then the exact cost of the next candle, so that the approximation and the exact figure can be set side by side.

Q11 starts from a price–demand equation. Revenue is the number sold times the price each one sells at, R(x) = x·p, with p written in terms of x first; profit is revenue minus cost, P(x) = R(x) − C(x). Differentiate each with the power rule. Part (b) is about reading signs: a positive marginal profit means one more mat sold adds to profit, a negative one means it takes away, and the size says roughly by how much. Every interpretation names the quantity, the level and the units.

The synthesis question

Q12 puts the set together on a ski-rental shop. Part (a) is Q11's construction: revenue from the price equation, then subtract the cost. Part (b) is Q5's method on a polynomial: write P(x) = 0, find its roots, place them on a number line together with the ends of the allowed interval, and test the sign on each piece. Part (c) is a sign chart for P′(x): find where marginal profit is zero, then read off where it is positive and where it is negative, and translate each sign into a sentence about one more rental.

Preview all 9 pages

Click any page to open the full PDF.

Getting the most out of it

Substitute before you do any algebra

Every limit starts with direct substitution. A number ends the question; 0/0 tells you to factor, rationalise or combine fractions; a nonzero number over zero tells you to look at the one-sided behaviour. Decide from the result, not from how the expression looks.

Say every result in the units of the problem

A limit of average cost is dollars per copy; a marginal cost is dollars per unit at a given level. Finish each business question with one sentence naming the quantity, the level and the units. That sentence is where most of the marks in this course sit.

Rewrite, then differentiate

Before using the power rule, put every term in the form c·xⁿ. Most errors in this set are a root or a denominator differentiated as it stood, not a slip in the rule itself.

Draw the number line

For continuity, inequalities and the sign of a marginal function, the same picture does the work: cut points in order, one sign per interval. Draw it every time; it is faster than reasoning in your head and easier to check.

Common questions